Kevin Mirallas' deflected strike put the Premier League club in front and Blackburn only had something to celebrate three minutes from time when Scott Dann netted a consolation.
Ross Barkley caught the eye with an impressive performance and set up the opening goal on 25 minutes with a neat piece of skill and blocked shot, which came back to Mirallas to fire home from 15 yards via a touch off Grant Hanley.
Roberto Martinez made seven substitutions at the break and Jelavic quickly made his mark in his first pre-season outing as he headed in a Steven Pienaar corner after 55 minutes.
The Croatian grabbed his second of the afternoon, and Everton's third, with 13 minutes remaining as he slammed in a shot with his left foot at the second attempt.
Blackburn had a decent spell at the end of the game and reduced their arrears when Dann headed home a Morten Gamst Pedersen corner three minutes from time.
Everton now head off on their tour of America and face Juventus in San Francisco on Wednesday.